Global log canonical thresholds of minimal $(1,2)$-surfaces

Let $S$ be a minimal surface of general type with $p_g(S)=2$ and $K_S^2=1$, so called by a minimal $(1,2)$-surface. Then we obtain that the global log canonical threshold of the surface $S$ via $K_S$ is greater than equal to $\frac{1}{2}$. As an application we have $$\mathrm{v}\mathrm{o}\mathrm{l}(X)\ge \frac{4}{3}{p}_g(X)-\frac{10}{3}$$ for all projective $3$-folds $X$ of general type which answers Question 1.4 of [J. A. Chen, M. Chen, C. Jiang, “The Noether inequality for algebraic threefolds”, arXiv:1803.05553] about Noether inequality for $X$ with $5\le p_g(X)\le 26$.